Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 31, 07 March 2024
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On Euclidean, spherical and hyperbolic crystallography
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Abstract
A major impetus in the early development of group theory in the 19th century was the study of geometrical symmetries. Inspired by the theory regarding orbifold notation developed by John Conway, we understand and analyze the theory of the classification of symmetrical patterns in three different spaces. One of the triumphs was the full understanding of 2-dimensional planar symmetries, precisely, the classification theorem of wallpaper groups. Then we use the same way to classify the symmetrical patterns in spherical and hyperbolic spaces. There were also scattered theories on the general notion called crystallographic groups. In this paper, we reproduce the classical result that there are 17 types of wallpaper groups using a topological method; we also conclude that there are 14 family cases in the spherical space and infinite conditions in the hyperbolic spaces. During this process, we first consider the three different spaces, then, analyze the symmetrical patterns case by case. The characteristic we consider to classify these patterns is orbifold; finally, we use the formulas to calculate the result.
Keywords
Symmetrical patterns, isometry groups, orbifold notation
References
1. Maxwell Levine. Plane symmetry groups. 2008.
2. J.H. Conway, H. Burgiel, and C. Goodman-Strauss. The Symmetries of Things. AK Peters/CRC Recreational Mathematics Series. Taylor & Francis, 2008.
3. H.S.M. Coxeter and W.O.J. Moser. Generators and Relations for Discrete Groups.
4. J. Gallier and D. Xu. A Guide to the Classification Theorem for Compact Surfaces. Geometry and Computing. Springer Berlin Heidelberg, 2013.
5. D. L. Johnson. Topics in the theory of group presentations: Examples of presentations. 1980.
6. J.H. Conway and D.A. Smith. On Quaternions and Octonions. Ak Peters Series. Taylor & Francis, 2003
Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
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- Volume Title
- Proceedings of the 3rd International Conference on Computing Innovation and Applied Physics
- ISBN (Print)
- 978-1-83558-317-3
- ISBN (Online)
- 978-1-83558-318-0
- Published Date
- 07 March 2024
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/31/20241107
- Copyright
- 07 March 2024
- Open Access
- This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited